A scaled complete-law protocol for comparing mediator organizations¶
The amplitude feasibility theorem admits a whole interval of interventions with a central organization-dependent contrast separated from the declared noisy null alternative. This protocol fixes one intervention and an independent forward calculation. It preserves the actual acquired source families while using nominal full-state trajectories as numerical references and transporting them with a separate, proved source-error bound.
This owner admits and freezes a prospective numerical policy. It does not evaluate a response or promise that the chosen interval algorithm will complete within its width target. The execution plan owns any subsequent execution and retention of its first outcome.
F1 and F2: fixed source, law, intervention and observation¶
Retain all 54 original form and continuous phase coordinates on three C9 rings, contacts \((4,13)\), \((13,22)\), unit capacities and the complete law
The degree matrix \(D\) is fixed by the actual joined support. Compare the two component classes \((1,1,1)\) and \((1,2,1)\). The original preparation and source costs remain unchanged: after the specified acquisition and relaxation, each component has separate zero-sum form and target-phase deviations with Euclidean norms at most \(\epsilon=10^{-32}\). The two class families have independent residuals. Within each class, all four continuations share one actual complete source.
Select
This choice follows a declared criterion, not a response search. On the already admitted interval \([4/3,7/5]\), use its uniform upper bound \(E(7/5)\) for the complete higher-amplitude and source allowance. The certified oriented lower bound \(-j_+\lambda^3-E(7/5)\), with the retained \(j_+<0\), is strictly increasing. The upper endpoint maximizes that certified bound while retaining every proved work and identity guard.
The two donor-only form jumps retain node \(4\), their exact times and all complete preevent states. Neither event changes phase. The missing-event controls are \((00,10,01,11)\); the delayed-only history carries its unprobed prefix. No source, mediator, mean, phase lift or hidden mode is reset at the second event.
For class \(k\), let
There are eight final scalar form readings. Each retains its own error bound \(\delta=10^{-30}\), with no distributional assumption. Hence \(|\widehat D_{\mathrm c}-D_{\mathrm c}|\le8\delta\). The separately noisy zero-class-contrast alternative has statistic in \([-8\delta,8\delta]\). Its exclusion through (3) requires the strict sufficient true-response condition beyond \(\pm16\delta\), not merely a signed numerical midpoint.
Keep radius \(r=1/12\), contact-work allowance \(10^{-12}\) and each probe-work allowance \(2\,10^{-6}\). The carried-work theorem uses the actual donor Laplacian in the delayed work and proves these guards at (2). It supplies no autonomous attachment or intervention selector. Forward smooth-domain flags are not substitutes for these source, work, mean-leaf and identity premises.
Actual source covers and numerical references¶
Let \(\Pi=[\pi_-,\pi_+]\) be the independently certified 256-bit rational enclosure of \(\pi\) supplied by the shared Machin construction. For component \(c\), local node \(j=0,\ldots,8\) and class tuple \(k_c\), retain the canonical actual-source covers
Use the correct endpoint order for negative phase coefficients and retain the exact rational pairs before dyadic materialization. The original componentwise Euclidean caps and zero sums describe a correlated acquired subset of these covers. Arbitrary independent corners need not be acquired or satisfy those correlations.
The independent full-law numerical references instead use
These boxes enclose the exact nominal target \(x=0,\ \theta_{c,j}=2\pi k_c(j-4)/9\). They are proof references for a numerical comparison, not instructions to replace or sharpen the physical preparation. The actual source families and their covers (4) are retained separately in the protocol association. Every reference history obeys the complete nonlinear 54-coordinate law and the same carried events as its actual counterpart. It is not a tangent trajectory or an amplitude-coefficient solution.
This separation addresses a numerical dependence issue without discarding physical source uncertainty. Direct propagation of Cartesian source uncertainty by interval arithmetic can lose the correlation between a source coordinate and its increment. Even the scalar contractive equation \(z'=-z\) illustrates the problem: direct interval Taylor terms from \(z(0)\in[-\epsilon,\epsilon]\), combined without their common-source dependence, enlarge that radius despite contraction of the exact flow. Finer steps do not in general remove this source wrapping. This example is a method warning, not a proof that the complete sine comparison or every box method must fail.
No assumed factorization of actual sources is introduced by (5). The following bound transports numerical reference observations to the unchanged acquired families after the forward calculation.
Independent full-law transport of actual-source errors¶
Set \(g=1/3000>\gamma\). For an actual history and its matching exact nominal reference, let \(X(t)\) and \(Y(t)\) be maximum-norm differences in the original form and phase coordinates. Initially both are at most \(\epsilon\). Heat contraction, the real normalized sine Lipschitz bound two and the complete phase row give
The same supplied jump is applied to both trajectories at each event, so their differences do not jump. Consequently
This bound is global in the real phase lifts. It does not divide phase uncertainty by \(\gamma\), assume nominal reflection symmetry of actual residuals, or replace the source family by a zero-error state.
If the two complete forward reference calls supply mixed intervals \(I_1,I_2\), form their independent cross-class enclosure
The factor eight counts the eight scalar endpoints. Independence of cross-class source residuals is retained; no actual-source cancellation is assumed. The same actual source within each four-history family remains essential to the definition of \(M_k\).
The transport claim applies to the original correlated family relative to its exact nominal target, not to every independent corner of the \(\Pi\)-expanded boxes (4). Target rounding and outward materialization in (5) already belong to \(I_{\mathrm{ref}}\); they are not ignored or represented as measurement noise. At the fixed design,
The coefficient prediction supplies neither \(I_1\), \(I_2\) nor the transport mechanism (6)-(8).
F3: complete forward execution and numerical budget¶
Use the existing four-history full-state producer once per class with (5). Admit both complete primitive sources and the common execution parameters before the first call. Fix class-one then class-two order. Retain each class's two prefixes and four suffixes, full-state endpoint handoffs and exact donor jumps. No cross-class prefix is shared.
For each class, retain all four raw endpoint bands and the primary mixed interval formed from the four suffix receiver increments. Shared-prefix receiver coordinates cancel symbolically within that class. This avoids subtracting an endpoint box from itself, but does not restore all remaining dependencies between steps or branches. Keep the raw endpoint cross-class difference as a separate diagnostic; neither construction is intersected with the analytic prediction.
Fix the following numerical policy before the reserved response:
| Quantity | Fixed value or rule |
|---|---|
| Common scale | \(7/5\), selected by the certified monotone lower bound |
| Form impulses | \(a=b=7/10000\) at structural times 0 and 1 |
| Final time | 2 |
| Step | \(1/128\), clipped only at declared event/final boundaries |
| Taylor order | 16 |
| Unique attempts | At most 768 per class; 1536 in total |
| Complete branch work | Two prefixes and four suffixes, each of 128 steps per class |
| Arithmetic | Shared exact rational and outward dyadic128 intervals |
| Target enclosure | Independent 256-bit rational \(\pi\) bounds, then outward source materialization |
| Tube and remainder | Unchanged strict Picard inclusion and source-box Taylor kernel |
| Width postcondition | \(\operatorname{width}(I_{\mathrm{actual}})\le10^{-30}\) |
| Adaptation | None: no step, precision, source, amplitude or coefficient search |
| First failure | Stop all later steps, branches and class calls; retain completed and failed evidence |
For a bounded order-17 remainder coefficient, halving the step doubles the number of steps and reduces the accumulated local truncation scale by \(2^{16}\). This motivates the selected step relative to the earlier one-class calculation, but it is not a proof of the actual interval width. Natural interval jets, Picard admission, arithmetic rounding and wrapping still require their own retained postconditions. No bound on exact derivatives alone is presented as a bound on the computed interval width.
The width target applies after the source transport (8). It leaves the reference calculation the exact allowance
Both widths and \(B_{\mathrm{source}}\) are retained separately. This is a fixed sufficient acceptance budget, not a guarantee that the selected algorithm meets it. Removing repeated propagation of \(\epsilon\) through independent boxes does not remove that obligation.
There is one shared cap of 1536 attempts. Each child receives the remaining allowance, subject to the producer's existing per-call cap; its fixed grid has at most 768 planned attempts and admits no retry. Thus the first child's retained cap need not equal the second's. The cap counts kernel attempts, including a failed attempt. A failure in class one leaves class two unattempted. A failure in class two retains the completed class-one result and available class-two prefixes. No missing history, final time or cross-class contrast is represented by zero. The reference and transported contrasts remain unavailable until both complete four-history calls succeed.
F4: prediction, sufficient resolution and first-outcome classification¶
Exact homogeneity of the retained complete cubic enclosure, with the proved fifth-and-higher amplitude remainder, supplies the nominal reference prediction
Adding the independent source allowance in (8) to the underlying exact rational nominal bound, before display rounding, gives the separately rounded actual-family prediction
These are open intervals with exact terminating rational endpoints. Equation (12) is not obtained by expanding the already rounded display endpoints in (11). They are fixed predictions; they never enter the forward field, Taylor coefficients, tube selection or endpoint arithmetic. The full comparison uses the forward \(I_{\mathrm{actual}}\), without intersecting it with (12) or selecting a favorable point from either interval.
The required width is tied to the predicted margin. For valid forward enclosures, a common true value belongs to \(I_{\mathrm{actual}}\cap J_{\mathrm{actual}}\). If (10)'s transported width postcondition holds, then
Thus that postcondition is sufficient for the desired separation, with more than \(2.157\,10^{-30}\) reserve. This conditional implication does not promise completion or width. The actual forward sign and separation inequalities are checked directly.
Retain distinct exact decisions:
- Forward availability: both class calls complete their full states and four observations under the declared attempt caps, law, source, clock, event and numerical evidence.
- Width: both reference and transported widths are reported; the latter meets its fixed \(10^{-30}\) budget.
- Consistency: the closed numerical reference interval overlaps (11), and the closed transported interval overlaps (12). For \(I=[l,u]\), \(J=(L,U)\), overlap means \(u>L\) and \(l<U\). Numerical-band containment in a prediction is a separate diagnostic, not required agreement or a reason to shrink it.
- True and recorded signs: respectively \(\sup I_{\mathrm{actual}}<0\) and \(\sup I_{\mathrm{actual}}<-8\delta\). The recorded band is \(I_{\mathrm{actual}}+[-8\delta,8\delta]\).
- Named-alternative exclusion: \(\sup I_{\mathrm{actual}}<-16\delta\) separates its recorded statistic from the independently noisy zero-contrast statistic. The raw eight-record sets are then disjoint through their fixed linear statistic.
Missing forward observations make dependent bounds and decisions unavailable. An available but wide interval can be inconclusive; failure of a sufficient test is not an overlap theorem. Disjoint forward and prediction intervals are conflicting evidence requiring an audit of numerical, mathematical and source premises, not an automatic physical counterexample. If discrimination inequalities pass but the width policy fails, retain both facts without claiming that every protocol postcondition passed.
Freeze source recipes, all realized reference and actual-cover bounds, the original acquired-family association, the law/support/clock, event order, observation, prediction, numerical policy, source code and stopping predicates before executing a reserved response. Preserve the first completed or failed outcome without changing parameters after seeing it. Subsequent correction or refinement would be separate evidence, not a rewrite of this protocol.
The result sought is a complete-law software comparison under declared source and observation budgets. It neither supplies a laboratory clock/noise calibration nor identifies a unique physical law, fundamental particle property or autonomous intervention.
Shared execution and retained-evidence boundaries¶
The two-source composition admits both primitive reference covers and runs the existing full-state producer in fixed order. Its evidence reconstruction checks consumed support, degrees, law, clock, source, events, all carried coordinates, Taylor arithmetic, observation identities and partial-failure associations before forming a cross-source contrast. Stored derivative enclosures and Picard generation remain explicit execution premises; arithmetic reconstruction is not a solver replay.
The protocol helper keeps canonical actual and reference covers separate and applies source transfer, sensor budgets and consistency decisions only after receiving an independently justified reference interval. It neither supplies a forward response nor validates acquisition from a cached verdict. Its primitive domain checks do not establish the external law and source hypotheses.
The composition controls and independent protocol algebra exercise source association, carried branch evidence, interval subtraction, source-error counts, strict open-boundary comparisons and conditional numerical resolution without executing the selected reference histories.
Immutable source and protocol association¶
The prospective text above is preserved unchanged. Its unevaluated statements describe protocol admission; the later outcome has a separate section below.
The exact protocol,
source archive
and freeze receipt
pin source commit 8e40dac013c026c0f198a2f95131512277286bbf without runtime
overlays. The archive contains the prospective text above, loaded runtime
owners and supplemental freeze/evaluation scripts; restoration starts from
the complete committed base. Its 1,433,580 bytes have SHA-256
564642f21fea44929f53463e2391a10e569bfe81b72e1c4e964eb3c5610424f7.
The receipt associates the exact source recipes, original acquired-family
proof, retained coefficient evidence, both predictions and all numerical,
source-transfer and recording criteria. Digests associate bytes without
authenticating chronology, execution or scientific premises.
The historical status is not_evaluated. No selected reference, coefficient,
acquisition, attempt or response was generated by this freeze. The archived
evaluator checks the pinned runtime versions and primitive policy before
creating an exclusive attempt. It preserves the first complete or partial
outcome, with a separate export-error record when serialization fails;
existing or partial outcome bytes are never replaced. Future response
evidence remains separate from these immutable files.
Inspect the association without importing or running archived code:
python scripts/restore_frozen_source.py --receipt docs/assets/sine_formed_classes/class-comparison-readout-v1.freeze.json
The same shared tool can restore the full pinned base into a new external worktree when the execution plan activates its reserved evaluation. The freeze audit checks this association and synthetic helper behavior without scientific execution. The sole next-task queue owns later evaluation and follow-up work. The numerical width was an unevaluated postcondition at freezing; its first outcome is retained below.
Retained first independent comparison¶
The single evaluation from the pinned complete source above completed all
1536 attempted steps: two classes, each with six carried segments of 128
steps. No evaluation error, source reset, retry, parameter change or
prediction intersection was used. Both the primary suffix-increment
contrast and the separate raw-endpoint contrast have retained status
discrimination_certified. The historical not_evaluated freeze status
is unchanged.
The primary reference interval is exactly
The raw-endpoint reference interval is \([-6549030940/2^{128},-6548962075/2^{128}]\). Their widths are respectively \(60535/2^{128}\) and \(68865/2^{128}\). Equation (8) adds the same independently proved source allowance once to each construction. The resulting bounds below are rounded outward; retained exact rational endpoints determine every decision.
| Quantity | Primary retained bound, in units of \(10^{-30}\) |
|---|---|
| Independent reference contrast | \([-19.245859943492,-19.245682047114]\) |
| Source allowance \(B_{\mathrm{source}}\) | \(\le0.080106809079\) |
| Actual-family contrast | \([-19.325966752570,-19.165575238035]\) |
| Reference width | \(\le0.000177896377\) |
| Actual-family width | \(\le0.160391514534<1\) |
| Recorded contrast, including eight reading errors | \([-27.325966752570,-11.165575238035]\) |
| Gap from the independently noisy zero-contrast alternative | \(>3.165575238035\) |
Both forward reference intervals lie strictly inside the open reference prediction (11); both transported intervals lie strictly inside the actual-family prediction (12). Thus the required overlap checks and the stronger, optional containment diagnostics pass. The actual-family width requirement passes for both constructions: the raw-endpoint width is at most \(0.160415994204\,10^{-30}\), with null-separation margin greater than \(3.165562998200\,10^{-30}\). The primary construction was selected prospectively; these are not alternative trials from which to choose a favorable response.
The true and recorded contrasts are strictly negative. In the fixed class order, this means \(M_1<M_2\): the two organizations have different nonlinear mixed responses. It is not a coherence or stability ranking. The recorded contrast is disjoint from \([-8\delta,8\delta]\), the declared independently noisy zero-class-contrast alternative. This finite separation uses the sixteen-error criterion, not merely a signed midpoint.
Byte association and reconstruction¶
The exclusive attempt
and compressed response
retain the first evaluation. The ZIP contains only
class-comparison-readout-v1.json, preserving its original bytes.
| Retained object | Bytes | SHA-256 |
|---|---|---|
| Attempt ledger | 723 | 0f55f35c32edde0cc5584767d84d03aaeb5113a87c55a8de9004b10c8d5077d7 |
| Response ZIP | 42,090,162 | c77d5b48a3e77e3e2c826dfb6c850dba73769d861ddc95acbc5bcf1828d8131c |
| Uncompressed response member | 808,798,262 | a6553e20450454a5ff191b608c35a651bd85a82c9ba9dfc5ed78c98302aff03b |
The read-only response audit re-admits source, support, law, clock, events and numerical policy. It uses the shared evidence reader to reconstruct all retained step arithmetic, full-state handoffs, both observation constructions and global completion. It then independently rebuilds source transport, widths, prediction checks and reading-error decisions. It executes no producer, field, derivative, Picard certificate generator or archived evaluator. Stored derivative and Picard generation remain numerical execution premises; reconstruction and digests do not authenticate execution or source acquisition.
This result is an independent complete-law numerical enclosure plus the proved transfer to the original correlated acquired families. No formation was rerun, and arbitrary Cartesian source corners are not thereby acquired. The work and identity claims retain the matched amplitude/source theorem; solver smoothness flags cannot replace it. The old unscaled central/spatial noise-cancellation results remain valid for their unchanged interventions.
The evaluation supports organization-dependent nonlinear interaction at one declared finite design. Repeated use must retain branch-specific state and memory; it cannot assume a new common source after each word. Laboratory preparation and reading precision, physical identification, autonomous intervention selection and inheritance across scales remain open.